Whakaoti i te 4/x+4/x-6=1

$$\frac { 4 } { x } + \frac { 4 } { x - 6 } = 1$$

Whakaoti mō x

$$x=2$$

$$x=12$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 0,6 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x-6\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x-6.

$$\left(x-6\right)\times 4+x\times 4=x\left(x-6\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-6 ki te 4.

$$4x-24+x\times 4=x\left(x-6\right)$$

Pahekotia te 4x me x\times 4, ka 8x.

$$8x-24=x\left(x-6\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x-6.

$$8x-24=x^{2}-6x$$

Tangohia te x^{2} mai i ngā taha e rua.

$$8x-24-x^{2}=-6x$$

Me tāpiri te 6x ki ngā taha e rua.

$$8x-24-x^{2}+6x=0$$

Pahekotia te 8x me 6x, ka 14x.

$$14x-24-x^{2}=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$-x^{2}+14x-24=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx-24. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=14 ab=-\left(-24\right)=24$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 24.

$$1,24 2,12 3,8 4,6$$

Tātaihia te tapeke mō ia takirua.

$$1+24=25 2+12=14 3+8=11 4+6=10$$

Ko te otinga te takirua ka hoatu i te tapeke 14.

$$a=12 b=2$$

Tuhia anō te -x^{2}+14x-24 hei \left(-x^{2}+12x\right)+\left(2x-24\right).

$$\left(-x^{2}+12x\right)+\left(2x-24\right)$$

Tauwehea te -x i te tuatahi me te 2 i te rōpū tuarua.

$$-x\left(x-12\right)+2\left(x-12\right)$$

Whakatauwehea atu te kīanga pātahi x-12 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-12\right)\left(-x+2\right)$$

Hei kimi otinga whārite, me whakaoti te x-12=0 me te -x+2=0.

$$x=12 x=2$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 0,6 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x-6\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x-6.

$$\left(x-6\right)\times 4+x\times 4=x\left(x-6\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-6 ki te 4.

$$4x-24+x\times 4=x\left(x-6\right)$$

Pahekotia te 4x me x\times 4, ka 8x.

$$8x-24=x\left(x-6\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x-6.

$$8x-24=x^{2}-6x$$

Tangohia te x^{2} mai i ngā taha e rua.

$$8x-24-x^{2}=-6x$$

Me tāpiri te 6x ki ngā taha e rua.

$$8x-24-x^{2}+6x=0$$

Pahekotia te 8x me 6x, ka 14x.

$$14x-24-x^{2}=0$$

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$-x^{2}+14x-24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 14 mō b, me -24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-1\right)\left(-24\right)}}{2\left(-1\right)}$$

Pūrua 14.

$$x=\frac{-14±\sqrt{196-4\left(-1\right)\left(-24\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-14±\sqrt{196+4\left(-24\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -24.

$$x=\frac{-14±\sqrt{196-96}}{2\left(-1\right)}$$

Tāpiri 196 ki te -96.

$$x=\frac{-14±\sqrt{100}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 100.

$$x=\frac{-14±10}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-14±10}{-2}$$

Nā, me whakaoti te whārite x=\frac{-14±10}{-2} ina he tāpiri te ±. Tāpiri -14 ki te 10.

$$x=-\frac{4}{-2}$$

Whakawehe -4 ki te -2.

$$x=2$$

Nā, me whakaoti te whārite x=\frac{-14±10}{-2} ina he tango te ±. Tango 10 mai i -14.

$$x=-\frac{24}{-2}$$

Whakawehe -24 ki te -2.

$$x=12$$

Kua oti te whārite te whakatau.

$$x=2 x=12$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 0,6 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x-6\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x-6.

$$\left(x-6\right)\times 4+x\times 4=x\left(x-6\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-6 ki te 4.

$$4x-24+x\times 4=x\left(x-6\right)$$

Pahekotia te 4x me x\times 4, ka 8x.

$$8x-24=x\left(x-6\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x-6.

$$8x-24=x^{2}-6x$$

Tangohia te x^{2} mai i ngā taha e rua.

$$8x-24-x^{2}=-6x$$

Me tāpiri te 6x ki ngā taha e rua.

$$8x-24-x^{2}+6x=0$$

Pahekotia te 8x me 6x, ka 14x.

$$14x-24-x^{2}=0$$

Me tāpiri te 24 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$14x-x^{2}=24$$

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$-x^{2}+14x=24$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+14x}{-1}=\frac{24}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\frac{14}{-1}x=\frac{24}{-1}$$

Whakawehe 14 ki te -1.

$$x^{2}-14x=\frac{24}{-1}$$

Whakawehe 24 ki te -1.

$$x^{2}-14x=-24$$

Whakawehea te -14, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -7. Nā, tāpiria te pūrua o te -7 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-14x+\left(-7\right)^{2}=-24+\left(-7\right)^{2}$$

Pūrua -7.

$$x^{2}-14x+49=-24+49$$

Tāpiri -24 ki te 49.

$$x^{2}-14x+49=25$$

Tauwehea x^{2}-14x+49. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7\right)^{2}=25$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{25}$$

Whakarūnātia.

$$x-7=5 x-7=-5$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$x=12 x=2$$